Stable topology and stable compactness

Abstract: To solve a large class of noncompact problems ubiquitous in random functional analysis and its applications, most of compactness-based fundamental theorems established for normed spaces or locally convex spaces are required to be extended for random normed modules and random locally convex modules in an appropriate way of weakening compactness. To achieve this goal, in this paper we first develop stable set theory by introducing some new set-theoretic tools, then propose a class of topological spaces with a stable structure, called stable topological spaces, and in particular propose and systematically develop the theory of stably compact spaces, which is a natural generalization of that of compact spaces. The resulting theory is of independent interest from the perspective of general topology since its investigation requires an analysis of stratification structures coordinated with partitions of unity in a complete Boolean algebra. Finally, we show the power of the theory of stable compactness by establishing random Banach-Alaoglu, random James and random Banach-Bourbaki-Kakutani-Šmulian theorems.